In this paper, we would like to prove the existence of global solutions to the FL equation on the line
from the perspective of inverse scattering transform.
This technique has been applied to prove the existence of global solutions to the derivative NLS equation
by Pelinovsky and Shimabukuro [1]}.
By establishing the Lipschitz continuity of the eigenfunctions and the scattering data,
we obtain the Lipschitz continuous mapping between the initial data and the solution to the FL equation.
In this way we are able to establish the existence of local and global solutions to the FL equation.
Compared with the derivative NLS equation, we find that extension of this approach to the FL equation will confronts some substantial difficulties,
which are also different from the derivative NLS equation [1]}.
